Rota-Baxter operators and relative Rota-Baxter operators in the context of Lie algebras first appeared in the study of classical \(r\) -matrices [1]}. Cohomologies of relative Rota-Baxter operators on Lie algebras were initially studied in [2]}. Then using higher derived brackets, the cohomology theory of relative Rota-Baxter Lie algebras were successfully established by Lazarev, Sheng and Tang in [3]}. Recently, cohomologies of relative Rota-Baxter Lie algebras with coefficients in an arbitrary representation were given by Jiang and Sheng in [4]}. Applications of these cohomologies were given to study infinitesimal deformations, abelian extensions and homotopies of relative Rota-Baxter Lie algebras (we refer to [5]}, [6]} for deformations and cohomologies of algebraic structures). Some aspects of the above study are already considered for relative Rota-Baxter (associative) algebras by the present authors [7]}. More precisely, we defined cohomologies of relative Rota-Baxter algebras, which can be used to characterize infinitesimal deformations. Homotopy relative Rota-Baxter algebras were also given in the same paper. However, cohomologies of relative Rota-Baxter algebras are not yet studied from the representation-theoretic points of view.
